Library of Congress Cataloging-in•Publicatlon Data Freeden. W. Multiscale potential theory: with applicatioll.'l to geoscience / Willi Freeden. Volker Michel. p. cm. -(Applied and numerical harmonic analysis) Includes bibliographical ~fe~l\Ces and index. ISBN-13: 978-1-4612-7395-0 I. Pmential theory (Mathematics) 2. Earth sciences.--Mathematics I. Michel, Volker. [I.

This paper provides an overview of two topics. First, it presents a unified approach to various techniques addressing the non-uniqueness of the solution of the inverse gravimetric problem; alternative, simple proofs of some known results are also given. Second, it summarises in a concise and self-contained way a particular multiscale regularisation technique involving scaling functions and wavelets.

The basic inverse problems for the functional imaging techniques of electroencephalography (EEG) and magnetoencephalography (MEG) consist in estimating the neuronal current in the brain from the measurement of the electric potential on the scalp and of the magnetic field outside the head. Here we present a rigorous derivation of the relevant formulae for a three-shell spherical model in the case of independent as well as simultaneous MEG and EEG measurements. Furthermore, we introduce an explicit and stable technique for the numerical implementation of these formulae via splines. Numerical examples are presented using the locations and the normal unit vectors of the real 102 magnetometers and 70 electrodes of the Elekta Neuromag (R) system. These results may have useful implications for the interpretation of the reconstructions obtained via the existing approaches.

The inverse problem of recovering the Earth's density distribution from data of the first or second derivative of the gravitational potential at satellite orbit height is discussed for a ball-shaped Earth. This problem is exponentially ill-posed. In this paper, a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part, the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG (satellite gravity gradiometry) satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust. Moreover, the noise sensitivity of the regularization technique is analysed numerically.

Row‐column designs allow error control in field experiments by blocking in two dimensions. While this strategy can capture spatial heterogeneity aligned with blocks and account for effects due to the farming operations along rows and columns, it suffers from the occasional clustered occurrence of several replications of the same treatment. This property of classical row‐column designs has hampered their more widespread use in practice. A further issue of practical importance is the degree of neighbor balance of a design, that is, the frequency of adjacencies of pairs of treatments. This paper proposes two design strategies to simultaneously optimize the evenness of spatial distribution of treatment replication as well as neighbor balance. Three examples are given to illustrate the proposed methods and demonstrate that both approaches yield comparable and satisfactory results.

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Sparse regularization has recently experienced high popularity in the inverse problems community. In this paper, we show that a sparse regularization technique can also be developed for linear geophysical tomography problems. For this purpose, we adapt a known matching pursuit algorithm. The main theoretical features (existence, stability, and convergence) of the new method are given. We also show further properties of some trial functions which we use. Moreover, the algorithm is applied to a static and a monthly varying gravitational field of South America which yields spatial and temporal variations in the mass distribution. The new approach represents essential progress in comparison to a corresponding wavelet method, which is not flexible enough for the use of heterogeneous data, and a respective spline method, where the resolution cannot exceed approximately 10 4 basis functions due to experienced numerical problems with the ill-conditioned and dense matrix. The novel sparse regularization technique does not require homogeneous data and is not limited in the number of basis functions due to its iterative algorithm.

Current activities and recent progress on constructive approximation and numerical analysis in physical geodesy are reported upon. Two major topics of interest are focused upon, namely trial systems for purposes of global and local approximation and methods for adequate geodetic application. A fundamental tool is an uncertainty principle, which gives appropriate bounds for the quanti®cation of space and momentum localization of trial functions. The essential outcome is a better understanding of constructive approximation in terms of radial basis functions such as splines and wavelets.